Homoclinic orbits to invariant sets of quasi-integrable

exact maps

Patrick Bernard

To cite this version:

Patrick Bernard. Homoclinic orbits to invariant sets of quasi-integrable exact maps. ErgodicTheory and Dynamical Systems, Cambridge University Press (CUP), 2000, 20 (6), pp.1583-1601. <hal-01251209>

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Homoclinic orbits to invariant sets of quasi-integrable exact

maps

Patrick Bernard

october 98

In Hamiltonian mechanics, various questions and results concern the becoming of theinvariant tori of an integrable system after perturbation. It is well known, by KAM theory,that many of these invariant tori are deformed but not destroyed. But it is known as wellthat many of them are destroyed. They give rise to more or less complicated invariant setssuch as Mather invariant sets [9].

One particular kind of tori that can be destroyed is resonant tori, that is invariant torifoliated by lower dimensional invariant tori. Treschev proved in [11] that if the frequencyinduced on the lower dimensional torus satisfies some Diophantine condition, then one of thelower dimensional tori is preserved, and becomes hyperbolic (whiskered in Arnold’s terminol-ogy [1]). If the unperturbed Hamiltonian is positive definite, it has been proved by Bolotin[3] that this preserved torus admits homoclinic orbits in the perturbed system. Similar re-sults have been obtained by Eliasson [5] for 1-resonant tori without the convexity assumption.These results are important not only as generalizations to high dimensional systems of oldresults on twist maps, but also because they are a step in the attempt to generalize Arnold’sconstruction of [1] and study diffusion in high dimensional Hamiltonian systems. Unfortu-nately, as is explained in [8], there are not enough preserved tori for Arnold’s constructionto be led directly. That’s why the case of resonant tori which do not satisfy Diophantineconditions and which can be completely destroyed is of some interest. Bolotin gives partialanswer to that question in [3], obtaining semi-asymptotic orbits to the Mather invariant setbut does not obtain an invariant set with homoclinic orbits.

In this paper, we focus our attention to a general resonant torus no lower dimensionaltorus of which has to be preserved. We prove that its destruction gives rise to compactinvariant sets, the Peierl’s sets introduced by Mather in [10] and containing the usual Mathersets, and that these sets admit nontrivial homoclinic orbits. As we stressed above, this resultcould have some interest in the study of diffusion, allowing to fill the gaps between preservedtori. Our method, that provides a different and less involved proof of Bolotin’s result, is basedon periodic orbits: both the invariant sets and the homoclinic orbits are obtained as limits ofperiodic orbits which are found by the variational methods of [2]. That the Peierl’s set is thenatural asymptotic set to be considered has just been noticed by Fathi [6] by quite differentmethods.

For convenience, we will consider exact maps instead of Hamiltonian flows, the linksbetween both theories are well known. Let us quote for instance [7] or [2] for details aboutconvexity assumptions. We also refer to [2] where it is explained how the local study of anyquasi-integrable exact map around a positive definite torus can be reduced to the followingsetting.

1

1 Main Result

We will consider an integrable symplectic diffeomorphism

Φ0 : Tn × Rn −→ Tn × Rn,

whose pull backφ0 : Rn × Rn −→ Rn × Rn

can be writtenφ0(x, y) = (x+ ω(y), y).

In all the following, we will suppose that Φ0 satisfies the following two hypothesis:

Hypothesis 1 (non-degeneracy) the map ω : Rn −→ Rn is a diffeomorphism.

Then φ0 admits a global generating function of type S [4], i.e. there exists a function S :Rn × Rn −→ R such that

(x1, y1) = φ0(x0, y0)⇐⇒{y0 = ∂1S(x0, x1)y1 = −∂2S(x0, x1)

. (1)

It is easy to see that in our case

S(x0, x1) = h(x1 − x0),

and we will make the following second assumption

Hypothesis 2 (convexity) the function h : Rn −→ R is strictly convex.

Let Φ be a C1 exact perturbation of Φ0, its pull back φ has a C2 generating function

S(x0, x1) = h(x1 − x0) + εP (x0, x1),

where the perturbation P satisfies

Hypothesis 3 (exactness) P (x0 +m,x1 +m) = P (x0, x1) for all m ∈ Zn.

Since we are interested in the dynamical structure of a bounded region of phase space, wecan change the mapping at infinity and furthermore suppose (see [2])

Hypothesis 4 (localization) there is a compact set K ⊂ Rn and a number a > 0 such that

S(x, x+ t) = h(t) + εP (x, x+ t) = a|t|2 + 0

when t /∈ K.

Let y0 and ω0 = ω(y0) be fixed. Assume that ω0 is resonant, which means that there existsk ∈ Zn such that < k, ω0 >∈ Z. Let

R = {k ∈ Znsuch that < k, ω0 >∈ Z}

and let r = rank (R), then we say that ω0 is r-resonant. The free group R has a basis B ofcardinal r. It is not hard to see that there is a foliation of the torus y = y0 in ergodic subtoriand a group morphism

F : Tn × Tn−r −→ Tn

2

such that these tori are {F (x, θ), θ ∈ Tn−r

}.

The unperturbed map restricted to {y = y0} is given by

F (x, θ) 7−→ F (x, θ) + ω0 = F (x, θ + ω0).

The frequency ω0 ∈ Rn−r is called the induced frequency. It is useful to define the averagedperturbation

P : Tn −→ R

given by

P(x) =

∫Tn−r

P(F (x, θ) , F (x, θ) + ω0

)dθ

= limN→+∞

1

2N

N∑i=−N

P(x+ iω0 , x+ (i+ 1)ω0

).

The function P is then constant along the ergodic subtori,

P(F (x, θ)) = P(x).

Hypothesis 5 P is minimal on exactly one of the ergodic tori {F (x, θ), θ ∈ Tn−r}, which isa non-degenerate critical manifold for P. We will note Γ0 this minimal torus.

We will use the notation π both for the canonical projection Rn −→ Tn and for (π, Id) :Rn×Rn −→ Tn×Rn and Π for the first factor projection Tn×Rn −→ Tn or Rn×Rn −→ Rn.We also set

Uδ = {q ∈ Tn, dist(q,Γ0) < δ},

and we will write Uδ for π−1(Uδ). We are now in a position to state our main result:

Theorem 1 Let Φε be the exact map whose pullback φε is generated by a C2 function

Sε(x0, x1) = h(x1 − x0)− εP (x0, x1)

satisfying H 1-5. Let C > 0 be any constant, then there are two nonnegative constants ∆and A and a function δ(ε) with limε→0 δ = 0 such that for any ε 6 ∆ and any c ∈ Rn with‖c−∇h(ω0)‖ 6 Cε, there exists a compact invariant set Σc(ε) of Φε satisfying

• Π|Σc(ε) is a bilipschitz homeomorphism onto its image.

• Σc(ε) ⊂{

(x, y) ∈ Tn × Rn/x ∈ Uδ(ε) , ‖y − y0‖ 6 A√ε},

• Σc(ε) has r + 1 homoclinic orbits lying in the zone{‖y − y0‖ 6 A

√ε}

of phase space.

• Both the invariant sets and the homoclinic orbits belong to Per(Φ).

The invariant set Σc(ε) is a Peierl’s set as defined in Mather [10], it will be described withsome details in section 3. If the induced frequency ω0 satisfies some Diophantine condition,and if the maps involved are sufficiently regular, Treschev [11] obtains the existence of anhyperbolic invariant torus of frequency ω0. In this case, theorem 1 becomes:

3

Theorem 2 (Bolotin,[3]) There is a constant C > 0 such that if there exists a KAM hy-perbolic torus Γ with rotation vector ω0, then there is c satisfying the hypothesis of theorem 1such that

Σc(ε) = Γ.

The torus Γ thus admits r + 1 homoclinics.

Remark : The existence of 2r homoclinic orbits may actually be proved in both statementsabove by noticing that the system is almost reversible in the region of phase space underinterest.

In Section 2, we introduce the variational setting for periodic orbits that will be used inall the following. In section 3, we describe the asymptotic set. We first state some generalproperties of Peierl’s sets, proving results of [10] in order to stress that no minimal measuresare needed to introduce the Peierl’s set. In the subsections we use averaging methods inspiredfrom [3] to localize the asymptotic set. In section 4, we explain that certain sequences ofperiodic orbits converge to homoclinic orbits, and in section 5, we use this fact to get severalnontrivial homoclinics.

2 Periodic orbits

All the constructions below will be based on periodic orbits, that will be obtained as actionminimizers. We will first introduce a variational problem for orbits of φ in the universal coverRn × Rn. Let

ETw = {(x0, . . . , xT ) ∈ (Rn)T+1such that xT = x0 + w},

Consider the LagrangianLTw : ETw −→ R

LTw(x0, . . . , xT ) =

T−1∑i=0

S(xi, xi+1).

It is not hard to see that there exists a T -periodic orbit((x0, y0), (x1, y1), . . . , (xT , yT )

)of φ if and only if

(x0, . . . , xT ) ∈ crit(LTw).

The orbit having a given x-projection is then uniquely defined, it satisfies:

yi = ∂1S(xi, xi+1) = −∂2S(xi−1, xi).

This variational setting has been introduced and studied in [2], we will need the followingpart of their result, in which only estimate (2) is non-trivial.

Theorem 3 (Bernstein Katok) Given C > 0, there exist two non negative constants ∆and A, depending on h and C but neither on w or T nor on P , such that if ‖dP‖C0 6 C andε 6 ∆, the above variational problem LTw admits a minimizing orbit X = (xi) satisfying

|xi+1 − xi − w/T | 6 A√ε. (2)

4

There are corresponding problems on Tn. For any c ∈ Rn, let

Sc(x0, x1) = S(x0, x1)− < c, x1 − x0 > .

The functional cLTw associated to Sc has the same critical points and the same minima than

that associated to S. We can now define

ET = {(q0, . . . , qT ) ∈ (Tn)T+1such that qT = q0}

and

HTc (q0, . . . , qT ) =

T−1∑i=0

Hc(qi, qi+1),

whereHc(q0, q1) = min

π(x0)=q0,π(x1)=q1(Sc(x0, x1))

In the following, we will usually use q for points and orbits in Tn and x for points andorbits in Rn. Given T and c, it is easy to see that the function

w ∈ Zn 7−→ min(cLTw)

is proper. It must have a minimum, and we will call W Tc the set of rotation numbers w ∈ Zn

realizing this minimum. If X is a minimizing orbit of LTw with w ∈W Tc then it is easy to see

that its projection Q ∈ ET is a minimizer of HTc , and any minimizer of HT

c can be obtainedthat way. The minimizing orbits of the functionals Hc depend on c, and if w, T and c arefixed, the projection Q ∈ ET of a minimizing orbit X ∈ ETw of LTw need not be a minimizingorbit of HT

c . Following Mather, we will call configurations the elements of ETw or ET or ingeneral all sequences of points. Let us end this section with an important remark:

Lemma 1 The functionHc : Tn × Tn −→ R

is Lipschitz continuous.

Proof: Let us first notice that there is a constant C depending on c such that for any (q, p) ∈(Tn)2 there are covering points (x, y) of (q, p) with ‖(x, y)‖ 6 C and Hc(q, p) = Sc(x, y). Letus now take two other points q′ and p′ and their covering points x′ and y′ closest to x and y.The result follows from the inequality

Hc(q′, p′)−Hc(q, p) 6 Sc(x

′, y′)− Sc(x, y) 6 K‖(x′, y′)− (x, y)‖ 6 Kd((q′, p′), (q, p)).

�

3 The asymptotic set

We will now define and describe the invariant set to which homoclinic orbits will be searched.Let

α(c) = infT>0

(minHT

c

T

).

This function is the α function of Mather. The following lemma will be of great importance:

5

Lemma 2lim infT−→∞

(min(HT

c − Tα(c)))

= 0.

Proof: Let us set mT = min(HTc −Tα(c)). If for some T we have mT = 0 then calling QT the

minimizing sequence of HTc − Tα(c) and n ∗ QT the nT -periodic configuration consisting of

n iterates of QT we have HnTc (n ∗QT )− nTα(c) = 0 for any n, and so lim inf mT = 0. Thus

if we assume that lim inf mT > 0, then there is a real number l satisfying

mT > l > 0.

Let us fix r > 0 such that |Hc(z, x) − Hc(z, y)| 6 l/2 when d(x, y) 6 2r. There is a realnumber p with 0 < p < 1 such that, given T points on the torus, there is a ball of radiusr containing at least pT of them. Take a minimizing configuration QT = (qTi ) of HT

c , afterpermutation of the indices, there are K > pT points qT0 , q

Ti1, . . . , qTiK−1

contained in a ball of

radius r. The configurations Qk = (qTik−1, qTik−1+1, . . . , q

Tik

) of length Tk = ik − ik−1 are almostperiodic and we have

HTkc (Qk)− Tkα(c) > l/2,

adding these inequalities for all k we get

HTc (Q)− Tα(c) > Kl/2 > pT l/2.

This givesα(c) 6 HT

c /T − pl/2

which is in contradiction with the above definition of α. �The following definition is that of [10], but we avoid the use of minimal measures.

Definition 1 A Peierl’s point is a point q ∈ Tn such that there exists a sequence Qn ofTn-periodic configurations with qn0 = q and

limn−→∞

HTnc (Qn)− Tnα(c) = 0.

The set of all Peierl’s points is the Peierl’s set Σc.

That this set is not empty is an easy consequence of lemma 2. Take a sequence Qn of Tn-periodic configurations such that limn−→∞H

Tnc (Qn) − Tnα(c) = 0. Any accumulation point

of the bounded sequence (qn0 )n∈N is a Peierl’s point. In order to get further information onthe set Σc, we will need the

Lemma 3 Let us fix q0 ∈ Tn and let Qn and Pn be two periodic configurations with qn0 =q0 = pn0 and

limn−→∞

(Hc − Tα)(Qn) = 0 = limn−→∞

(Hc − Tα)(Pn),

then qn1 and pn1 have the same limit q1.

Proof: We first take subsequences so that qn1 , pn1 , qnTnq −1 and pnTnp −1 have limits q1, p1, q−1 and

p−1. Consider the configurations

Rn = (qn0 , qn1 , . . . , q

nTnq −1, q, p

n1 , . . . , p

nTnp

)

6

andRn(q) = (qn0 , q

n1 , . . . , q

nTnq −1, q0, p

n1 , . . . , p

nTnp

), q ∈ Tn,

and lethn(q) = Hc(R

n(q))− Tnα(c),

where Tn = Tnp + Tnq . We then have

hn(q) = cn +Hc(qnTnq −1, q) +Hc(q, p

n1 ).

Since hn(q0) −→ 0 the sequence cn has a limit −Hc(q−1, q0)−Hc(q0, p1). The function hn(q)thus has a limit h(q) which satisfies h(q0) = 0 and h > 0. It follows that the function

q 7−→ Hc(q−1, q) +Hc(q, p1)

must have a minimum at q0. Taking points x−1, x0 and y1 in Rn covering q−1, q0 and p1 andsuch that Hc(q−1, q0) = Sc(x−1, x0) and Hc(q0, p1) = Sc(x0, x1), we obtain that the function

x 7−→ Sc(x−1, x) + Sc(x, y1)

must have a minimum at x0 and the equation

∂2Sc(x−1, x0) = −∂1Sc(x0, y1)

holds true. Let us take a given sequence Qn such that qn−1 has a limit q−1, then any accumula-tion point of pn1 for any admissible sequence Pn must have a covering that satisfies the aboveequation hence the sequence pn1 must have a limit, which corresponds to the only solution ofthe equation. We get that p1 = q1 by applying this to the case Pn = Qn. �

The mappingΦc : Σc −→ Σc

q 7−→ q1.

is Lipschitz continuous. We could prove it using arguments as in lemma 3 but more involved,we prefer to send the reader to [9]. The set

Σc = {(q, ∂1Sc(q,Φc(q))) , q ∈ Σc}

is Φ-invariant. There is a map φc : π−1(Σc) −→ π−1(Σc) so that the following diagramcommutes.

π−1(Σc)φ //

��

π

{{xxxx

xxxx

xπ−1(Σc)

��

π

{{xxxx

xxxx

x

ΣcΦ //

Π

��

Σc

Π

��

π−1(Σc)φc //

π

{{vvvvvvvvvπ−1(Σc)

π

{{vvvvvvvvv

ΣcΦc // Σc

7

The vertical projections Π on this diagram are all bilipschitz homeomorphisms. We will oftenuse the notation Σc for π−1(Σc) and Σc for π−1(Σc). We have given the general definition of aPeierl’s set. In order to obtain asymptotic orbits to it, some information must be obtained onits topology. We now focus our attention to the vicinity of the torus Tn×{y0}. The followingproposition provides us with some Peierl’s set in this zone of phase space, and gives us thetopological information we need.

Proposition 1 Let us fix C > 0, there is a function δ(ε) with limε−→0 δ(ε) = 0, and aconstant A > 0 such that when ε is small enough we have

Σc ⊂ Uδ(ε)

andΣc ∈ {‖y − y0‖ 6 A

√ε} ∈ Tn × Rn

for any c such that ‖c−∇h(ω0)‖ 6 Cε.

In the following subsections, we prove proposition 1.

3.1 Averaging

Let us setS(x0, x1) = S(x0, x1) + f(x0)− f(x1),

where f ∈ C1(Tn). All the functionals defined above on sets of periodic orbits are unchangedwhen S is replaced by S. We will use this fact to reduce our functionals to more convenientforms.

Lemma 4 For any o > 0, There is a function fo ∈ C1(Tn), such that the modified generatingfunction

So(x0, x1) = S(x0, x1) + εfo(x0)− εfo(x1)

satisfiesSo(x0, x1) = h(x1 − x0) + εP(x0) + εR(x0, x1)

with|R(x0, x0 + ω0)| 6 o.

Proof: We first formally solve the equation

P (x, x+ ω0) + f(x)− f(x+ ω0) = P(x0).

Let us put

P (x, x+ ω0) =∑k∈Zn

pkei<k,x>

thenP(x) =

∑k∈R

pkei<k,x>

and the function f(x) =∑

k∈Zn fkei<k,x>, with

fk =pk

ei<k,ω0> − 1, k /∈ R ; fk = 0 , k ∈ R

8

formally solves the equation. Under our hypothesis, the above series can be divergent. Nev-ertheless, given o > 0, there is K ∈ R such that∣∣∣∣∣∣

∑|k|>K

pkei<k,x>

∣∣∣∣∣∣ 6 o.It is then easy to see that

fo(x) =∑|k|6K

fkei<k,x>

satisfies‖P (x, x+ ω0) + fo(x)− fo(x+ ω0)− P(x)‖∞ 6 o.

this proves lemma 4. �It will be useful in the following to use the notations

Sc = Sc − α(c), Hc = Hc − Tα(c),

andQ(x) = P(x)−min(P).

Lemma 5 Let us fix C > 0, there are non negative numbers a and b and a function K(o)independent of ε such that for any c ∈ Rn satisfying ‖∇h(ω0)− c‖ 6 Cε,

a‖t‖2 + εQ(x0)− 4oε−Kε2 6 cSo(x0, x1) 6 b‖t‖2 + εQ(x0) + 4oε+Kε2, (3)

where t = x1 − x0 − ω0.

Proof: For ε small enough, the function x1 7−→ h(x1 − x0) + εR(x0, x1) is uniformly strictlyconvex thus there exists a > 0 such that

h(x1 − x0) + εR(x0, x1) >

h(ω0) + εR(x0, x0 + ω0)+ < ∇h(ω0) + ε∂2R(x0, x0 + ω0), x1 − x0 − ω0 > +a‖x1 − x0 − ω0‖2.

This gives

cSo(x0, x1) > h(ω0)− < c, ω0 > −εo+ < ∇h(ω0) + ε∂2R(x0, x0 + ω0)− c, t >+a‖t‖2 + εP(x0)

> h(ω0)− < c, ω0 > −εo− εC‖t‖+ a‖t‖2 + εP(x0)

> h(ω0)− < c, ω0 > −εo−Kε2 + a‖t‖2 + εP(x0)

changing the constant a in the last inequation. We can also obtain the majoration

cSo(x0, x1) 6 h(ω0)− < c, ω0 > +εo+Kε2 + b‖t‖2 + εP(x0).

We are now in a position to estimate α(c). The above minoration gives

α(c) > h(ω0)− < c, ω0 > +εminP − εo−Kε2

> h(ω0)− < c, ω0 > +εminP − 2εo

9

for ε small enough with respect to o. Let x and y be two points of Γ0, consider the configurationxk = x+ kω0, let T be the time between ε−1/2 and 2ε−1/2 for which xT is closest to y and setd = y − xT . Now consider the configuration Y = (yk = xk + kd/T ), it connects x and y andsatisfies

cLo(Y ) 6 bT−1‖d‖2 + Tεmin(P) + Th(ω0)− T < c, ω0 > +εoT + ε2KT

because

cSo(yk, yk+1) 6 b‖d/T‖2 + εP(yk) + h(ω0)− < c, ω0 > +oε+Kε2 + h(ω0).

Since d goes to zero as ε goes to zero, we obtain for ε small enough the estimate

cLo(Y ) 6 Tεmin(P) + Th(ω0) + T < c, ω0 > +2o√ε.

If we apply this remark to the case x = y, Y is a periodic configuration and we get

α(c) 6 h(ω0)− < c, ω0 > +εminP + 2oε.

We have estimated α(c):

|α(c)− h(ω0)+ < c, ω0 > −εminP| 6 2oε.

The inequality (3) is obtained by mixing this estimate with the above inequalities. �We have also proved:

Lemma 6 Let x and y be two points of Γ0, there is a configuration X = (x = x0, . . . , xT = y)with ε−1/2 6 T 6 2ε−1/2 and

cLo(X) 6 10o√ε.

3.2 Localization of the asymptotic set

In this rather technical section, we prove proposition 1. In the following calculations, we willmainly use inequality (3), thus our result will be true for any c satisfying the hypothesis oflemma 5. We will omit the subscript c.

Lemma 7 Let ν > 0 and x and y be two points of Uν , for o > 0 small enough and ε smallenough with respect to α, there is a configuration X = (x = x0, . . . , xT = y) satisfying

Lo(X) 6 Cν2√ε.

Proof : We will construct our connecting configuration via Γ0. Let x ∈ Uν , x+d its projectionon Γ0. Considering the configuration (x0, . . . , xT ) = (x, x + ω0 + d/T, . . . , x + Tω0 + d), weget

So(xi, xi+1) 6 b ‖d/T‖2 + εQ(x+ i(ω0 + d/T )) + εo+Kε2.

Adding these inequalities gives

Lo(X) 6 bν2/T + Tε

(supUν

Q+ o+Kε

),

10

and sinceP(x) 6 minP + C(d(x,Γ0))2,

L(X) 6 bν2/T + Tε(Cν2 + 4o+Kε

).

Taking T =√ε we obtain that

Lα(x0, . . . , xT ) 6 Cν2√ε.

In order to prove the lemma, we just have to consider a three part configuration

x = x0, . . . , xN , . . . , xM , . . . , xT = y,

where xN ∈ Γ0 and x0, . . . , xN is the orbit considered above, xM ∈ Γ0 and xM , . . . , xT is areversed configuration like above, and xN , . . . , xM is a configuration obtained by lemma 6.The inequality of the lemma is then satisfied if o is small enough. �

Corollary 1 Let x and y lie in Uν , any configuration connecting x and y satisfies

Lα > −Cν2√ε.

Proof: Let us consider a configuration connecting x and y, and complete it by the configurationof low action obtained above connecting y and x. We obtain a periodic configuration, writingthat its action must be non negative gives the inequality of the corollary. �Let 0 < ν < δ, and let W = Rn − Uν .

Lemma 8 Let X = x0, . . . , xT be a configuration with x0 ∈ Uν , xi ∈ W for i > 1 andxT ∈ Rn − Uδ. Then for α and ε small enough

Lα(X) > D√εν(δ − ν).

The conclusion holds true for a configuration with xT ∈ Uν , xi ∈ W for i < T and x0 ∈Rn − Uδ.

Proof: We will give the proof of the first part of the lemma, the proof of the second part goesalong the same line and is easier. Since Γ0 is a nondegenerate minimal manifold for P wehave

P(x) > minP +D(d(x,Γ0))2.

It follows thatSo(xi, xi+1) > a‖ti‖2 + εCν2 − 4oε−Kε2

for i > 1. We then choose o so that Dν2 − 4o > Dν2/2 > 0, and obtain

So(xi, xi+1) > C‖ti‖ν√ε,

for ε small enough and i > 1. On the other hand,

So(x0, x1) > a‖t0‖2 + εQ(x0)− 4oε−Kε2,

So(x0, x1) > a‖t0‖2 + εDν2 − εC‖t0‖ − 4oε−Kε2.

For ε small enough, we getSo(x0, x1) > D‖t0‖ν

√ε,

11

Adding all the minorations we have obtained gives the lemma because

∑i

‖ti‖ >

∥∥∥∥∥∑i

ti

∥∥∥∥∥ > |δ − ν|.�

Lemma 9 There is a function δ(ε) such that limε→0 δ(ε) = 0 and a function a(ε) satisfyinga(ε) > 0 for ε > 0 such that any periodic configuration (x0, x1, . . . x0 + m) of any rotationnumber leaving Uδ(ε), satisfies L(x0, . . . , x0 +m) > 2a(ε):

L(x0, . . . , x0 +m) > 2a(ε) if xi /∈ Uδ(ε) for some i.

Proof: Let δ be fixed, and let X be a periodic configuration leaving Uδ, let ν = δ2. If X staysout of Uν , then adding the inequalities

So(xi, xi+1) > a‖ti‖2 + εDν2 − 4oε−Kε2,

we obtainL(X) = Lo(X) > εDν2 − 4oε−Kε2,

for any small o, so that we haveL(X) > Dεδ4

for ε small enough. Let us now suppose that X comes into Uν . We can find a configurationY = x−i, . . . , x0, . . . , xj in X with x0 /∈ Uδ ,xk ∈W if −i < k < j and (x−i, xj) ∈ (Uν)2. Theabove lemma gives us

Lα(Y ) > D√εδ3.

By corollary 3, we then haveL(X) >

√ε(Dδ3 − Cδ4)

when ε is small enough with respect to delta. As a consequence of these results, there is aconstant D > 0 such that the inequality

L(X) > Dmin(√εδ3, εδ4

)must be satisfied when δ is small enough and when ε is small enough with respect to δ. Thelemma follows. �The horizontal localization of proposition 1 follows. Let us now end the proof of proposition 1:We are still given a fixed c satisfying the hypothesis of theorem 1, and we omit all subscripts c.Let Xk ∈ ETkwk be a sequence of periodic configurations such that L(Xk) −→ 0 and Tk −→∞.

Lemma 10 The rotation number wk satisfies:

‖wk/Tk − ω0‖ 6 2A√ε

when k large enough and ε small enough, where A is the constant of inequality (2).

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Proof: By inequality (2), we have

‖xki+1 − xki − wk/Tk‖ 6 A√ε.

Assume that ‖wk/Tk−ω0‖ > 2A√ε with the same constant A, then setting tki = xki+1−xki −ω0

as usual, we have that ‖tki ‖ > A√ε. Putting this in inequality 3 gives

L(Xk) > Tk(aAε− oε−Kε2)

for any o, which gives a contradiction when o and ε are small enough because Tk −→∞ andL(Xk) −→ 0. �Using lemma 10 and (2) we obtain

‖xki+1 − xki − ω0‖ 6 3A√ε

for k large enough. Vertical localization of proposition 1 follows, with a different constant A.

�

4 Convergence

In this section we will study sequences of periodic orbits, and obtain homoclinic orbits asaccumulation points. We fix c ∈ BCε(∇(ω0)), and we will omit the subscript c in the followingLet Qk be a sequence of Tk-periodic orbits such that H(Qk) −→ 0 and Tk −→∞. The orbitQk has a covering Xk ∈ ETkwk minimizing LTkwk . We can take a subsequence to be reduced tothe case where the inequality of lemma 10 is satisfied for every k, and where the sequencewk/Tk has a limit ω, which satisfies

‖ω − ω0‖ 6 2A√ε.

Let Xk(m) ∈ ETkwk+m be the minimizing sequence of LTkwk+m, and Qk(m) its associated orbiton Tn. We need some informations about these orbits.

Lemma 11 There is a constant C(m), depending on m but not on k, such that

L(Xk(m)) 6 C(m).

Proof:

L(Xk(m)) 6 L(xk0, . . . , xkTk−1, x

kTk

+m)

6 L(Xk) + S(xkTk−1, xkTk

+m)− S(xkTk−1, xkTk

)

6 L(Xk)−min S + sup‖x‖61,‖y‖61+m+ω0+3A

√ε

S(x, y).

�

Lemma 12 There is a constant C(m) such that for any 0 6 i < j 6 Tk

H(qki (m), . . . , qkj (m)) 6 C(m).

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Proof: Let us set P = (pj , . . . , pi) = (qkj (m), . . . , qkTk(m), qk1 (m), . . . , qki (m)). Writing

H(P ) + H(qki (m), qkj (m)) > 0

givesH(P ) > −max

(Tn)2H.

We get

H(qki (m), . . . , qkj (m)) = H(Qk(m))− H(P )

6 C(m) + max H.

�Thanks to these lemmas, we are in a position to study the accumulation points of the sequenceQk(m). Let us fix a sequence ak of integers, we can consider the centered orbit

Θk =(θki = qki+ak(m)

)i∈Z

,

where everything is defined by periodic continuation. There is a subsequence Θl(k) such thatl 7−→ θli has a limit θi for every i. The biinfinite sequence Θ = (θi)i∈Z will be called anaccumulation point of the centered orbit Θk. An accumulation point of a centered orbit ofa sequence Y k of periodic orbits will be called an accumulation point of Y k. We have thefollowing proposition, in which as usual everything depends on a vector c ∈ Rn satisfying‖c−∇h(ω0)‖ 6 Cε.

Proposition 2 Any accumulation point of Qk(m) is the projection of an orbit that is homo-clinic to Σ, any orbit obtained that way satisfies the localization ‖y − y0‖ 6 A

√ε of theorem

1.

This proposition provides no existence result up to now since the accumulation point couldbe an orbit contained in Σ.Proof: Since Θk are orbit segments, we have

∂2S(θki−1, θki ) = −∂1S(θki , θ

ki+1),

taking the limit we obtain∂2S(θi−1, θi) = −∂1S(θi, θi+1),

which means that θi is the projection of the orbit (θi, ∂1S(θi, θi+1)). Since ‖xki+1(m)−xki (m)−ω0‖ 6 4A

√ε, this orbit must stay in the zone ‖y − y0‖ 6 A

√ε (where A is a new constant).

To prove that this orbit is homoclinic, it is enough to prove that the α and ω-limit of theorbit (θi, ∂1S(θi, θi+1)) lie in Σ. We will prove it for the ω-limit, the other part being similar.It is sufficient to see that for any subsequence θj(i) having a limit θ, we have θ ∈ Σc andθj+1 −→ φc(θ). Taking a subsequence if necessary, we can suppose that d(θj , θ) 6 2−j . Letus consider the configuration

Zj = (θ, θj(i)+1, . . . , θj(i+1)−1, θ),

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it satisfies H(Zj) > 0 and

H(Zj) 6 H(θj(i), θj(i)+1, . . . , θj(i+1)) + C2−j .

Adding these inequalities and using lemma 11 yields

i1∑i=i0

H(Zj(i)) 6 C2−j(i0) + H(θj(i0), . . . , θj(i1+1))

6 C2−j(i0) + supk

(H(θkj(i0), . . . , θ

kj(i1+1))

)6 C2−j(i0) + C(m).

We obtain that H(Zi) −→ 0 so that θ ∈ Σc and θj(i)+1 −→ φc(θ). The last assertion fol-lows from the fact that φc(θ) is the only possible accumulation point of the sequence θj(i)+1. �

5 Nontriviality and multiplicity

As we noticed it in the previous section, proposition 2 does not guarantee the existence ofnontrivial homoclinic orbits. Some additional work will be needed. Let us fix δ small, thereis ε0 such that Σ ∈ Uδ when ε 6 ε0. There also exists η such that d(q,Σ) 6 η ⇒ q ∈ Uδ whenε 6 ε0. Let us define the open set

Vη ⊂ Rn × Rn = {(x0, x1) such that d ((x0, x1), graph(φc)) < η} ⊂ Uδ × Uδ.

We associate to any periodic orbit X ∈ ETw its relative homology

(hb(X) =< w, b > −T < ω0, b >)b∈B ∈ Zr,

recall that B is a basis of the resonant group. We also associate to any point x ∈ Uδ coefficients

(hb(x)) = E(< x, b >)b∈B ∈ Zr,

where E(x) is the integer closest to x, the above giving a good definition if δ has been chosensmall enough. These coefficients have of course no topological meaning.

Lemma 13 If δ has been chosen small enough, for any point x ∈ Σ,we have

hb(φc(x)) = hb(x)+ < ω0, b >,

from which easily follows that

(x0, x1) ∈ Vη ⇒ hb(x0) = hb(x1)+ < ω0, b > .

Proof: Let us consider a sequence Xk as above with L(Xk) −→ 0 and xk0 −→ x. Thesequence xk1 then has a limit φc(x) satisfying ‖φc(x) − x − ω0‖ 6 A

√ε. We obtain that

| < φc(x), b > − < x, b > − < ω0, b > | is small. Since both φc(x) and x are in Uδ,< φc(x), b > and < x, b > are close to integers, the result follows. �

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As an easy corollary of that lemma, we obtain that if Xk is an orbit such that L(Xk) −→ 0then (xki , x

ki+1) ∈ Vη for all i when k large enough, thus hb(X

k) =< xkTk , b > − < xk0, b >

−Tk < ω0, b > is close to hb(xkTk

)− hb(xk0)− Tk < ω0, b >= 0, so hb(Xk) = 0, which writes

< wk, b >=< ω0, b >

for k large enough. We also define the relative homology of a φ-orbit homoclinic to Σ asfollows,

hb(Θ) = limi−→∞

(hb(θi)− hb(θ−i)− 2i < ω0, b >).

This is well defined because for i large enough, we have θi ∈ Uδ and (θi, θi+1) ∈ Vη. We cannow prove the existence of nontrivial orbits. Let us fix some m and consider the sequenceXk(m). There is an alternative1- There exist ak such that

(xkak(m), xkak+1(m)

)admits an accumulation point out of Vη, and

so there is a nontrivial homoclinic, that leaves Uη.2- (xki (m), xki+1(m)) ∈ Vη holds when k is large enough.If 2- holds, it follows easily that the periodic orbit Xk(m) has trivial relative homology:

hb(Xk(m)) = 0.

This last equation writes< m, b >= 0,

we can choose m so that this equation is not satisfied, in this case 2- is impossible and so 1-is true.

We will now go in some kind of concentration compactness for our sequences Xk(m) toobtain multiplicity. Assume m has been chosen so that a nontrivial accumulation orbit leavingUη exists. There are a sequence ak0, an homoclinic orbit Θ0 and a subsequence of Xk(m) thatwill still be noted Xk(m) such that xk

i+ak0(m) −→ θ0

i for all i. Possibly changing ak0, we can

suppose that there exists A0 such that (θ0i , θ

0i+1) ∈ Vη for i < 0 or i > A0, and given ν > 0,

we can take a subsequence of our subsequence such that d(xki+ak0

(m), θ0i ) 6 ν for |i| 6 2A0,

this subsequence is still noted Xk(m). Let us define

ak1 = min{i > ak0 +A0 such that (xki (m), xki+1(m)) /∈ Vη}

andak−1 = max{i < ak0 such that (xki (m), xki+1(m)) /∈ Vη}.

We can extract another subsequence such that there exists a nontrivial homoclinic orbit Θ1,and an integer A1 satisfying

(θ1i , θ

1i+1) ∈ Vη when i > A1

andd(xk

i+ak1(m), θ1

i ) 6 ν when |i| 6 2A1.

We can continue the process by defining

akl = min{i > akl−1 +Al−1 such that (xki (m), xki+1(m)) /∈ Vη}

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and finding Θl and Al with(θli, θ

li+1) ∈ Vη when i > Al

andd(xk

i+akl(m), θli) 6 ν when |i| 6 2Al.

We can do the same for l < 0. Assume now that the process ends on each side, that is wehave l+ > 0 and l− 6 0 such that akl++1 > Tk and akl−−1 < 0 when k > k0, then we have

found a subsequence Xk(m) such that

d(xki (m), θli−akl

) 6 ν when akl − 2Al 6 i 6 akl + 2Al

and

(xki (m), xki+1(m)) ∈ Vη when akl +Al 6 i < akl+1 or Tk > i > akl+ +Al+

or 0 6 i < akl− .

Looking precisely at the construction we just performed and applying lemma 13, we see that

< m, b >= hb(Xk(m)) =

l+∑l=l−

hb(Θl). (4)

Lemma 14 If there are only a finite number N of geometrically distinct homoclinics thenthe process described above ends for each m. It follows that the relative homologies of thesehomoclinics must additively generate Zr and thus

N > r + 1.

Proof: If the process always ends, then the above sum formula (4) must be satisfied, and sincethe function

m 7−→ (< m, b >)b∈B ∈ Zr

is easily seen to be onto, we have the second part of the lemma. It remains to prove that theprocess ends. If there are only N homoclinic orbits, there are at most N leaving Uη, one ofthem must occur at least (l+ − l−)/N times in the sequence Θl defined above, that is thereare indices l1, . . . , lK with K > (l+ − l−)/N such that Θlj = Θ. Let us set

Y kj =

(xkaklj, xk

aklj+1, . . . , xk

aklj+1

).

and

Y kj =

(θ0, x

kaklj

+1, . . . , xk

aklj+1−1, θ0

).

The configuration Y kj is periodic and leaves Uη, so there must be o > 0 such that

L(Y kj ) > 2o.

If ν has been chosen small enough, it follows that

L(Y kj ) > o.

This givesC(m) > L(Xk(m)) > Ko

which is a majoration for K and thus for l+ − l−. �Acknowledgments : I thank Professor E. Sere for his comments and encouragement. I

thank the referee for useful remarks.

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References

[1] Arnold V. I. : Instability of Dynamical systems with several degrees of freedom, SovietMath. 5 (1964), 581-585.

[2] Bernstein D. - Katok A. : Birkhoff periodic orbits for small perturbations of completelyintegrable Hamiltonian systems with convex Hamiltonian, Invent. Math. 88(1987), 227-241.

[3] Bolotin S.V. : Homoclinic orbits to invariant tori of Hamiltonian systems, A. M. S.Translations, ser. 2 ,168 (1995), 21-90.

[4] Mc Duff D. - Salamon D. : Introduction to symplectic topology, Oxford Math. Monographs(1995).

[5] L. H. Eliasson : Biasymptotic Solutions of Perturbed Integrable Hamiltonian Systems,Bol. Soc. Brasil Mat. (N.S.) 25 (1994), 56-76.

[6] Fathi A. : Orbites heteroclines et ensembles de Peierls, CRAS 326 (1998), 1213-1216.

[7] Lochak P. : Canonical perturbation theory via simultaneous approximation, Russ. Math.Surveys 47 (1992) 57-133.

[8] Lochak P. : Arnold’s diffusion : a compendium of remarks and questions, in Proceedingsof the Barcelona 1995 conference on dynamical systems with more than 3 degrees offreedom, Kluwer, (1996).

[9] Mather J. : Action minimizing invariant measures for positive definite Lagrangian sys-tems, Math. Z. 207 (1991), 169-207.

[10] Mather J. :Variational construction of connecting orbits, Ann. Inst. Fourier, 43 (1993),1349-1368.

[11] Treschev D. : The mechanism of destruction of resonant tori of Hamiltonian systems,Math Sb, (1989), in English in Math. USSR Sbornik 68 (1991), 181-203.

Patrick BernardInstitut Fourier,Universite Grenoble I, BP 74, 38402 Saint Martin d’Heres cedex, France.Patrick.Bernard@ujf-grenoble.fr

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